Advertisements
Advertisements
प्रश्न
By the principle of mathematical induction, prove the following:
52n – 1 is divisible by 24, for all n ∈ N.
Advertisements
उत्तर
Let P(n) be the proposition that 52n – 1 is divisible by 24.
For n = 1, P(1) is: 52 – 1 = 25 – 1 = 24, 24 is divisible by 24.
Assume that P(k) is true.
i.e., 52k – 1 is divisible by 24
Let 52k – 1 = 24m
To prove P(k + 1) is true.
i.e., to prove `5^(2(k+1)) - 1` is divisible by 24.
P(k): 52k – 1 is divisible by 24.
P(k + 1) = `5^(2(k+1)) - 1`
= 52k . 52 – 1
= 52k (25) – 1
= 52k (24 + 1) – 1
= 24 . 52k + 52k – 1
= 24 . 52k + 24m
= 24 [52k + 24]
which is divisible by 24 ⇒ P(k + 1) is also true.
Hence by mathematical induction, P(n) is true for all values n ∈ N.
APPEARS IN
संबंधित प्रश्न
By the principle of mathematical induction, prove the following:
1 + 4 + 7 + ……. + (3n – 2) = `("n"(3"n" - 1))/2` for all n ∈ N.
By the principle of mathematical induction, prove the following:
32n – 1 is divisible by 8, for all n ∈ N.
By the principle of mathematical induction, prove the following:
n(n + 1) (n + 2) is divisible by 6, for all n ∈ N.
By the principle of mathematical induction, prove that, for n ≥ 1
13 + 23 + 33 + ... + n3 = `(("n"("n" + 1))/2)^2`
By the principle of Mathematical induction, prove that, for n ≥ 1
1.2 + 2.3 + 3.4 + ... + n.(n + 1) = `("n"("n" + 1)("n" + 2))/3`
Using the Mathematical induction, show that for any natural number n,
`1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/((3"n" - 1)(3"n" + 2)) = "n"/(6"n" + 4)`
Using the Mathematical induction, show that for any natural number n, x2n − y2n is divisible by x + y
Use induction to prove that 5n+1 + 4 × 6n when divided by 20 leaves a remainder 9, for all natural numbers n
Choose the correct alternative:
In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways
Choose the correct alternative:
Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is ______
